Embedded newform 177.2.e.a.7.3

• Label: 177.2.e.a.7.3
• Level: $177$
• Weight: $2$
• Character: 177.7
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			7.3
Character	\(\chi\)	\(=\)	177.7
Dual form			177.2.e.a.76.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.372741 - 0.549752i) q^{2} +(0.0541389 - 0.998533i) q^{3} +(0.576985 - 1.44812i) q^{4} +(2.31671 - 1.07182i) q^{5} +(-0.569126 + 0.342432i) q^{6} +(-0.243434 + 0.876769i) q^{7} +(-2.30852 + 0.508144i) q^{8} +(-0.994138 - 0.108119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{9}{29}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.372741	−	0.549752i	−0.263568	−	0.388734i	0.672760	−	0.739861i	\(-0.265109\pi\)
							−0.936328	+	0.351127i	\(0.885798\pi\)
\(3\)	0.0541389	−	0.998533i	0.0312571	−	0.576504i
							\(5\)	2.31671	−	1.07182i	1.03606	−	0.479335i	0.173262	−	0.984876i	\(-0.444569\pi\)
0.862802	+	0.505541i	\(0.168707\pi\)
\(7\)	−0.243434	+	0.876769i	−0.0920093	+	0.331387i	−0.995201	−	0.0978494i	\(-0.968804\pi\)
							0.903192	+	0.429237i	\(0.141217\pi\)
\(11\)	−3.02688	+	2.86721i	−0.912638	+	0.864496i	−0.991195	−	0.132413i	\(-0.957727\pi\)
							0.0785570	+	0.996910i	\(0.474969\pi\)
\(13\)	4.42728	−	0.481495i	1.22791	−	0.133543i	0.528916	−	0.848674i	\(-0.322599\pi\)
							0.698990	+	0.715132i	\(0.253633\pi\)
\(17\)	−1.03841	−	3.74003i	−0.251853	−	0.907091i	−0.976059	−	0.217507i	\(-0.930207\pi\)
							0.724206	−	0.689583i	\(-0.242206\pi\)
\(19\)	2.39487	−	1.82053i	0.549420	−	0.417658i	−0.293292	−	0.956023i	\(-0.594751\pi\)
							0.842712	+	0.538365i	\(0.180958\pi\)
\(23\)	−3.29168	+	6.20877i	−0.686363	+	1.29462i	0.258138	+	0.966108i	\(0.416891\pi\)
							−0.944501	+	0.328509i	\(0.893454\pi\)
\(29\)	0.798933	−	1.17834i	0.148358	−	0.218812i	−0.746247	−	0.665669i	\(-0.768146\pi\)
							0.894605	+	0.446857i	\(0.147457\pi\)
\(31\)	8.51654	+	6.47411i	1.52962	+	1.16278i	0.932794	+	0.360410i	\(0.117363\pi\)
							0.596822	+	0.802374i	\(0.296430\pi\)
\(37\)	−3.02084	−	0.664936i	−0.496622	−	0.109315i	−0.0404053	−	0.999183i	\(-0.512865\pi\)
							−0.456217	+	0.889869i	\(0.650796\pi\)
\(41\)	−0.0318184	−	0.0600158i	−0.00496920	−	0.00937290i	0.881015	−	0.473088i	\(-0.156861\pi\)
							−0.885984	+	0.463715i	\(0.846516\pi\)
\(43\)	−5.79940	−	5.49349i	−0.884401	−	0.837749i	0.103212	−	0.994659i	\(-0.467088\pi\)
							−0.987613	+	0.156910i	\(0.949847\pi\)
\(47\)	8.63465	+	3.99482i	1.25949	+	0.582704i	0.932106	−	0.362186i	\(-0.117970\pi\)
							0.327388	+	0.944890i	\(0.393832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.7.3	✓	140
3.2	odd	2		531.2.i.c.361.3		140
59.17	even	29	inner	177.2.e.a.76.3	yes	140
177.17	odd	58		531.2.i.c.253.3		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.7.3	✓	140	1.1	even	1	trivial
177.2.e.a.76.3	yes	140	59.17	even	29	inner
531.2.i.c.253.3		140	177.17	odd	58
531.2.i.c.361.3		140	3.2	odd	2